# Rule base and adaptive fuzzy operators cooperative learning of Mamdani fuzzy systems with multi-objective genetic algorithms

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## Abstract

In this paper, we present an evolutionary multi-objective learning model achieving cooperation between the rule base and the adaptive fuzzy operators of the inference system in order to obtain simpler, more compact and still accurate linguistic fuzzy models by learning fuzzy inference adaptive operators together with rules. The multi-objective evolutionary algorithm proposed generates a set of fuzzy rule based systems with different trade-offs between interpretability and accuracy, allowing the designers to select the one that involves the most suitable balance for the desired application. We develop an experimental study testing our approach with some variants on nine real-world regression datasets finding the advantages of cooperative compared to sequential models, as well as multi-objective compared with single-objective models. The study is elaborated comparing different approaches by applying non-parametric statistical tests for pair-wise. Results confirm the usefulness of the proposed approach.

## Keywords

Linguistic fuzzy modelling Interpretability-accuracy trade-off Multi-objective genetic algorithms Adaptive inference system Adaptive defuzzification Rule learning## Notes

### Acknowledgments

Paper supported in part by the Spanish Ministry of Education and Science under grant no. TIN2008-06681-C06-06, and the Andalusian government under grant no. P07-TIC-03179.

## References

- 1.Alcalá R, Alcala-Fdez J, Gacto MJ, Herrera F (2007) A multi-objective evolutionary algorithm for rule selection and tuning on fuzzy rule-based systems. In: Proceedings of Fuzz-IEEE’07 international conference on fuzzy systems. London, UK, pp 1367–1372Google Scholar
- 2.Alcalá R, Gacto MJ, Herrera F, Alcala-Fdez J (2007) A multi-objective genetic algorithm for tuning and rule selection to obtain accurate and compact linguistic fuzzy rule-based systems. Int J Uncertain Fuzzin Knowl Based Syst 15(5):539–557zbMATHCrossRefGoogle Scholar
- 3.Alcalá-Fdez J, Herrera F, Márquez F, Peregrín A (2007) Increasing fuzzy rules cooperation based on evolutionary adaptive inference systems. Int J Intell Syst 22(9):1035–1064zbMATHCrossRefGoogle Scholar
- 4.Alcalá-Fdez J, Sánchez L, Garcia S, Del Jesus M, Ventura S, Garrell J, Otero J, Romero C, Bacardit J, Rivas V, Fernandez J, Herrera F (2009) Keel: a software tool to assess evolutionary algorithms to data mining problems. Soft Comput 13(3):307–318CrossRefGoogle Scholar
- 5.Buckley JJ, Hayashi Y (1994) Can approximate reasoning be consistent? Fuzzy Sets and Syst. 65(1):13–18zbMATHCrossRefMathSciNetGoogle Scholar
- 6.Branke J, Kaubler T, Schmeck H (2000) Guiding multi-objective evolutionary algorithms towards interesting region. Technical Report No. 399, Institute AIFB, University of Karlsruhe, GermanyGoogle Scholar
- 7.Casillas J, Cordón O, Herrera F (2002) COR: a methodology to improve ad hoc data-driven linguistic rule learning methods by inducing cooperation among rules. IEEE Trans Syst Man Cybern 32(4):526–537Google Scholar
- 8.Casillas J, Cordón O, Herrera F, Magdalena L (eds) (2003) Interpretability issues in fuzzy modelling. Studies in fuzziness and soft computing. Springer, Heidelberg, p 128Google Scholar
- 9.Casillas J, Cordón O, Herrera F, Magdalena L (eds) (2003) Accuracy improvements in linguistic fuzzy modelling. Studies in fuziness and soft computing. Springer, Heidelberg, p 129Google Scholar
- 10.Casillas J, Cordón O, Del Jesus MJ, Herrera F (2005) Genetic tuning of fuzzy rule deep structures preserving interpretability for linguistic modelling. IEEE Trans Fuzzy Syst 13(1):13–29CrossRefGoogle Scholar
- 11.Cococcioni M, Ducange P, Lazzerini B, Marcelloni F (2007) A pareto-based multi-objective evolutionary approach to the identification of Mamdani fuzzy systems. Soft Comput 11:1013–1031CrossRefGoogle Scholar
- 12.Cordón O, Herrera F, Márquez FA, Peregrín A (2004) A study on the evolutionary adaptive defuzzification methods in fuzzy modelling. Int J Hybrid Intell Syst 1(1):36–48Google Scholar
- 13.Deb K, Agrawal S, Pratab A, Meyarivan T (2002) A fast and elitist multi-objective genetic algorithm: NSGA-II. IEEE Trans Evol Comput 6(2):182–197CrossRefGoogle Scholar
- 14.Demˇsar J (2006) Statistical comparisons of classifiers over multiple data sets. J Mach Learn Res 7:1–30MathSciNetGoogle Scholar
- 15.Eshelman LJ (1991) The CHC adaptive search algorithm: how to have safe search when engaging in nontraditional genetic recombination. Found Genet Algorithms 1:265–283Google Scholar
- 16.Eshelman LJ, Schaffer JD (1993) Real-coded genetic algorithms and interval-schemata. Found Genetic Algorithms 2:187–202Google Scholar
- 17.Gacto MJ, Alcalá R, Herrera F (2009) Adaptation and application of multi-objective evolutionary algorithms for rule reduction and parameter tuning of fuzzy rule-based systems. Soft Comput 13:419–436CrossRefGoogle Scholar
- 18.García S, Herrera F (2008) An extension on “statistical comparisons of classifiers over multiple data sets” for all pairwise comparisons. J Mach Learn Res 9:2579–2596Google Scholar
- 19.García S, Fernández A, Luengo J, Herrera F (2009) A study of statistical techniques and performance measures for genetics-based machine learning: accuracy and interpretability. Soft Comput 13(10):959–977CrossRefGoogle Scholar
- 20.García S, Molina D, Lozano M, Herrera F (2009b) A study on the use of non-parametric tests for analyzing the evolutionary algorithms’ behaviour: a case study on the CEC’2005 special session on real parameter optimization. J Heuristics (in press) doi: 10.1007/s10732-008-9080-4
- 21.Herrera F, Lozano M, Verdegay JL (1998) A learning process for fuzzy control rules using genetic algorithms. Fuzzy Sets and Systems 100:143–158CrossRefGoogle Scholar
- 22.Ishibuchi H, Murata T, Turksen IB (1997) Single-objective and two objective genetic algorithms for selecting linguistic rules for pattern classification problems. Fuzzy Sets Syst 89(2):135–150CrossRefGoogle Scholar
- 23.Ishibuchi H, Nozaki K, Yamamoto N, Tanaka H (1995) Selecting fuzzy if–then rules for classification problems using genetic algorithms. IEEE Trans Fuzzy Syst 3(3):260–270CrossRefGoogle Scholar
- 24.Ishibuchi H, Yamamoto T (2004) Fuzzy rule selection by multi-objective genetic local search algorithms and rule evaluation measures in data mining. Fuzzy Sets Syst 141(1):59–88zbMATHCrossRefMathSciNetGoogle Scholar
- 25.Krone A, Krause H, Slawinski T (2000) A new rule reduction method for finding interpretable and small rule bases in high dimensional search spaces. In: Proceedings of the 9th IEEE international conference on fuzzy systems. San Antonio, TX, USA, pp 693–699Google Scholar
- 26.Krone A, Taeger H (2001) Data-based fuzzy rule test for fuzzy modelling. Fuzzy Sets Syst 123(3):343–358zbMATHCrossRefMathSciNetGoogle Scholar
- 27.Márquez AA, Márquez FA, Peregrín A (2008) Cooperation between the inference system and the rule base by using multi-objective genetic algorithms. In: Proceedings of third international of hybrid artificial intelligence system, (HAIS’08). Burgos, Spain, pp 739–746Google Scholar
- 28.Márquez FA, Peregrín A, Herrera F (2007) Cooperative evolutionary learning of fuzzy rules and parametric aggregation connectors for Mamdani linguistic fuzzy systems. IEEE Trans Fuzzy Syst 15(6):1162–1178CrossRefGoogle Scholar
- 29.Narukawata K, Nojima Y, Ishibuchi H (2005) Modification of evolutionary multi-objective optimization algorithms for multi-objective design of fuzzy rule-based classification systems. In: Proceedings of the 2005 IEEE international conference on fuzzy systems. Reno, pp 809–814Google Scholar
- 30.Sheskin DJ (2003) Handbook of parametric and nonparametric statistical procedures. CRC Press, Boca RatonGoogle Scholar
- 31.Wang LX, Mendel JM (1992) Generating fuzzy rules by learning from examples. IEEE Trans Syst Man Cybern 22(6):1414–1427CrossRefMathSciNetGoogle Scholar
- 32.Wilcoxon F (1945) Individual comparisons by ranking methods. Biometrics 1:80–83CrossRefGoogle Scholar
- 33.Zhou SM, Gan JQ (2008) Low-level interpretability and high-level interpretability: a unified view of data-driven interpretable fuzzy system modelling. Fuzzy Sets Syst 159(23):3091–3131CrossRefMathSciNetGoogle Scholar
- 34.Zitzler E, Laumanns M, Thiele L (2001) SPEA2: improving the strength pareto evolutionary algorithm for multi-objective optimization. In: Evolutionary methods for design, optimization and control with applications to industrial problems (EUROGEN’01), pp 95–100Google Scholar